Assume $u \in L^\infty ([0,T];H^{s-1})$, $u_t \in L^\infty ([0,T];H^{s-1})$ then can we show that $u \in C([0,T];H^{s-1})$

Question

In Sogge's "Lectures on nonlinear wave equations" it states that :Assume $u \in L^\infty ([0,T];H^{s-1})$, $u_t \in L^\infty ([0,T];H^{s-1})$ then $u \in C([0,T];H^{s-1})$ .
Here $u_t$ is defined as a distribution s.t. $$<u_t,\phi(x,t)>=-\int_0^t \int_{R^n} u(x,s) \phi_t(x,s)\, dxds-\int_{R^n} u(x,0) \phi(x,0) \,dx$$ for $\phi \in C_0^{\infty}((-\infty,T) \times R^n)$.

Since $u \in H^1([0,T];H^{s-1})$ , if we can apply sobolev embedding theorem , then we have $C^0([0,T]) \subset H^1([0,T])$ . Can we just apply sobolev embedding theorem in this abstract sobolev spaces ? Can anyone suggest some text that gives a complete/thorough treatment of this kind of theorem ？Any help would be very appreciated.